Table Of ContentNon-standard Schwinger fermionic representation of unitary group
Fu-Lin Zhang and Jing-Ling Chen∗
Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R.China
(Dated: February 2, 2008)
The non-standard Schwinger fermionic representation of the unitary group is studied by using
n-fermion operators. One finds that the Schwinger fermionic representation of the U(n) group is
not unique when n ≥ 3. In general, based on n-fermion operators, the non-standard Schwinger
fermionic representation of theU(n) group can beestablished inauniform approach,whereall the
generators commute with the total number operators. The Schwinger fermionic representation of
U(Cnm) group is also discussed.
PACSnumbers: 03.65.-w;03.65.Fd;02.20.Sv
8
0
0 I. INTRODUCTION 3-fermion is not unique, by providing a non-standard
2
Schwingerfermionicrepresentation(NSSFR).InSec. IV,
n we develop a uniform approach to establish the NSSFR
Therearemany kinds ofimportantmappingsbetween
a ofthe U(n)groupbasedonn-fermionoperators(n 3),
spin systems and multi-boson or multi-fermion systems.
J ≥
whereallthe generatorscommute withthe totalnumber
They not only essentially simplify the analysis of the
9 operators. InSec. V,wediscusstheSchwingerfermionic
problems under consideration, but also help us to un-
] derstand various aspects of them. The most famous representationof the U(Cnm) groupbasedon n fermions.
h amongthesemappingsaretheSchwinger[1,2],Holstein- Conclusion is made in the last section.
p
Primakoff [3] bosonic representations and the Jordan-
-
t Wigner transformations [4]. The first two map an arbi-
n II. BRIEF REVIEW OF SSR
traryspintoabosonicsystem,whilethethirdestblishesa
a
u connectionbetweenone-dimensionalspin-1/2latticeand
q spinless formions on the same lattice. These representa- In the standard Schwinger bonsonic representation of
[ tions are very successful in describing magnetism in var- SU(2) group, the three generators are mapped onto the
iousquantumsystems[5,6,7,8,9,10]. Theyalsoplaya bilinear form of the bosonic operators as
1
significantroleinmanyothercontexts. Forexample,the
v
9 Schwingerrepresentationhas been exploitedinquantum Jb = 1(b†b +b†b )
5 optics [11] andin the study ofcertainclassesofpartially 1 2 1 2 2 1
3 coherent optical beams [12]. The Jordan-Wigner trans- 1
1 formation has also been used to simulate the interacting J2b = 2i(b†1b2−b†2b1) (1)
.
1 fermions with quantum computers [13]. 1
Jb = (b†b b†b ),
0 A lot of work has been done to generalize these map- 3 2 1 1− 2 2
8 pings in several ways. Chaturvedi and Mukunda have
0 extendedtheSchwingerrepresentationtotheSU(3)case whereb andb† (i=1,2)aretheannihilationandcreation
: i i
v on the primes of that each unitary irreducible represen- operators of the i-th boson, respectively.
i tation appears exactly once [14]. Kim has discussed the The commutation relations of n independent bosonic
X
Schwingerrepresentationbasedonthe mixedsets ofcre- operators are
r
a ationandannihilationoperatorsofbosonicandfermionic
type [15]. The Jordan-Wigner transformation has been [b ,b†]=δ , [b ,b ]=[b†,b†]=0. (2)
i j ij i j i j
generalized to arbitrary spin case and also the fermions
were replaced by anyons [16, 17]. In addition, there are Let Qb =b†b , it is easy to have
some new kinds of constructions of the SU(2) algebra ij i j
being reported [18, 19].
[Qb ,Qb ]=δ Qb δ Qb , (3)
The purpose of this Brief Report is to study the ij kl jk il− li kj
Schwinger representation of unitary group by using n-
whichremarkablysharesthe same commutationrelation
fermion operators. The work is organized as follows: In
ofmatricese ,having1inthe(i,j)position(i.e.,thei-th
Sec. II, to make the report be self-contained, we make ij
arrowandthej-thcolumnofthematric)and0elsewhere.
a brief review for the standard Schwinger representation
IfG isamatrixgeneratorofaLiegroupandGαβ denotes
(SSR)ofunitarygroupwithbosonsandfermions. InSec. i i
itselementsinthe(α,β)position,thentheoperatorsGb,
III, we show that the realization of the U(3) group with i
which are defined by the linear combinations of Qb as
ij
Gb = GαβQb = b†Gαβb , (4)
i X i αβ X α i β
∗Email:[email protected] αβ αβ
2
obeythesamecommutationrelationsasthoseofthema- j =1 representations of the U(2) algebra are
trices G . It means that the operators Gb form a rep-
i i
resentation of the Lie group represented by G . When 0 √2 0
n = 2 and G = σ /2 (where σ are the Pauli mi atrices, J(1) = J(1)+iJ(1) =0 0 √2,
i i i + 1 2
i=1,2,3),Eq. (4)givesthestandardSchwingerbosonic 0 0 0
representation of the U(2) group as in Eq. (1).
0 0 0
The annihilation and creation operators in Eq. (4) J(1) = J(1) iJ(1) =√2 0 0, (8)
− 1 − 2
are not necessarily restricted to bosons. Notice that the 0 √2 0
anticommutationrelationsofnindependentfermionsare
1 0 0
J(1) = 0 0 0 .
{ai,a†j}=δij, {ai,aj}={a†i,a†j}=0, (5) 3 0 0 −1
TheGell-Mannmatricescanbeconstructedbyusingthe
where ai and a†i are annihilation and creation operators quadratic forms of the above spin-1 matrices as
of the i-th fermion. The key point is that the bilinear
form operators Qfij = a†iaj also obey the commutation λ = √2(J(1)J(1)+J(1)J(1)),
relations of e . Thus replacing Qb by Qf in Eq. (4), 1 2 3 + − 3
ij αβ αβ
onegetsthestandardSchwingerfermionicrepresentation i√2 (1) (1) (1) (1)
λ = − (J J J J ),
of the Lie group 2 2 3 + − − 3
1
λ = [J(1)J(1),J(1)J(1)],
Gf = GαβQf = a†Gαβa . (6) 3 2 3 + − 3
i Xαβ i αβ Xαβ α i β λ4 = 21(J+(1)2+J−(1)2),
λ = −i(J(1)2 J(1)2), (9)
There are n2 independent operators Qb or Qf for n 5 2 + − −
ij ij
bosons or fermions. They can construct the U(n) Lie λ = −√2(J(1)J(1)+J(1)J(1)),
algebra, and all the generators commute with the to- 6 2 3 − + 3
Ntalf n=umPbni=er1Nopifer=atPorni=N1ba†i=ai.PSinin=c1eNtihbe=HaPminil=t1onb†iiabni,soorf λ7 = i√22(J3(1)J−(1)−J+(1)J3(1)),
tHhbe =n-dNimbe+nsin2o,naHlfbo=sonNicfa+ndn2,feIrtmiinondiiccaotescsiltlhataotrsboatrhe λ8 = 4√13[J+(1)2,J−(1)2]+ 2√13[J3(1)J−(1),J+(1)J3(1)].
n-dimensionalbosonic and fermionic oscillatorshave the
U(n) symmetry. Conseuently, if the spin-1 matrices in Eq. (9) are re-
placed by their standard Schwinger fermionic represen-
tations
Jf = √2(a†a +a†a ),
III. NSSFR OF U(3) GROUP WITH 3-FERMION + 1 2 2 3
Jf = √2(a†a +a†a ), (10)
− 2 1 3 2
Jf = a†a a†a =N N ,
In this section we would like to study the NSSFR of 3 1 1− 3 3 1− 3
U(3)groupwiththreefermions. Forthreefermions,there
ahigher-orderrepresentationoftheU(3)groupcanthen
is only one independent 6-order operator a†a a†a a†a
1 1 2 2 3 3 be obtained as
and no higher-order operator. The 2-order operators
have been used to realized the standard Schwinger λh = (a†a +a†a )(1 N )+(a†a +a†a )N ,
1 1 2 2 1 − 3 2 3 3 2 1
fermion representation. Here we attend to apply the 4- λh = ( ia†a +ia†a )(1 N )+( ia†a +ia†a )N ,
order operators to establish the NSSFR. 2 − 1 2 2 1 − 3 − 2 3 3 2 1
λh = N N 2N N +N N +N N ,
Inthefundamentalrepresentation[20]oftheU(3),the 3 1− 2− 1 3 1 2 2 3
generators are Gi = λi/2, i = 1,2,...,8, where λi are λh4 = (a†1a3+a†3a1)(1−2N2),
the Gell-Mann matrices. They obey the U(3) algebraic λh = ( ia†a +ia†a )(1 2N ), (11)
commutation relations 5 − 1 3 3 1 − 2
λh = (a†a +a†a )(1 N )+(a†a +a†a )N ,
6 2 3 3 2 − 1 1 2 2 1 3
8 λh = ( ia†a +ia†a )(1 N )+( ia†a +ia†a )N ,
7 − 2 3 3 2 − 1 − 1 2 2 1 3
[λ ,λ ]=2i f λ , (7)
i j X ijk k λh = (N +N 2N +2N N N N N N )/√3.
k=1 8 1 2− 3 1 3− 2 3− 1 2
One may verify that the commutation relations of λh
i
where f is the structure constant. Notice that the are the same as in Eq. (7) due to Eq. (5). All of the
ijk
3
operators λh commute with the total number operator. f(m)(m) = 1, and f(m)(x) = 0 when x equals to any
i n n
Itisworthytomentionthatthenon-standardSchwinger other integer between 1 and n 1. Then the NSSFR of
−
representation in Eq. (11) does not valid for bosons. U(3) in Eq. (11) can be recast to a very simple form
n=3
λh = [a†λ αβa f(1)(N)
IV. NSSFR WITH n-FERMION i X α i β 3
α,β=1
TheabovemethodofgettingtheNSSFRofU(3)group +a†αλ′iαβaβf3(2)(N)], (17)
is not conveniently applicable for the arbitrary U(n)
group. Thus we would like to rewrite Eq. (11) into a whereN =Pni=1a†iai denotes the totalparticlenumber,
uniformform,sothatitcanbedirectlygeneralizedtoan andf(1)(N)= N+2andf(2)(N)=N 1respectively.
arbitrary U(n) group. Bas3ed on Eq−. (17), the NS3SFR of U(n−) group can be
Letuslookatthecorrespondingmatrix-representation directly obtaioned as
ofEq. (11)inthe occupationnumber space,whosestan-
dard basis reads n
λh = [a†λ αβa f(1)(N)
i X α i β n
1, a†, a†, a†, a†a†, a†a†, a†a†, a†a†a† vac , (12) α,β=1
{ 1 2 3 1 2 1 3 2 3 1 2 3}| i
+a†λ′αβa f(n−1)(N)], (18)
α i β n
where vac is the vacuum state. Based on Eq. (12),
| i
the operators in Eq. (11) correspond the following 8 8 where λ′ = U( λ∗)U†, the matrix elements of U are
partitioned matrices: × U i =( −1)mi+1 and the others are zeros.
m,n+1−m
−
0 0 0 0
λh =0 λi 0 0, (13) V. REPRESENTATION OF U(Cnm) GROUP
i 0 0 λ 0 WITH n-FERMION
i
0 0 0 0
Inthen-fermionoccupationnumberspace,thedimen-
where λi denotes the i-th Gell-Mann matrix of U(3). sionofthesubspacewithN =misCm =n!/m!(n m)!.
Evidently, matrices λh satisfy the commutation rela- n −
i In such a subspace we can construct the Schwinger
tions of U(3) algebra because matrices λi do so. Sim- fermionicrepresentationofanU(Cm)group. Thelargest
n
iλlafrl=y, the3,3standaar†dλSαchβwaingcoerrrfeesrpmoniodnitcorethperesfoenlltoawtiinong onewecanconstructisU(Cn[n2])group,where[n2]denotes
i Pα=1,β=1 α i β the integer part of n/2.
matrices
Let us take the 4-fermion case as an example.
The subspace with the total number N = 2 is
0 0 0 0
λf =0 λi 0 0, (14) {a†1a†2,a†1a†3,a†1a†4,a†2a†3,a†2a†4,a†3a†4}|vaci, where a†i is the
i 0 0 λ′ 0 i-th fermionic creation operator. We introduce the no-
i
0 0 0 0 tations as O1 = a2a1,O2 = a3a1,O3 = a4a1,O4 =
a a ,O = a a ,O = a a and i = O† vac . Then
3 2 5 4 2 6 4 3 | i i| i
awnhderUe λd′ien=otUe(s−thλe∗i)uUn†i,taλr∗iyims tahtreixconjugate matrix of λi, tthhee soupbesrpaatocresaQreij, which behave as the matrices eij, in
0 0 1 Qij =|iihj|=Oi†|vacihvac|Oj =Oi†Ojf4(2)(N), (19)
U = 0 1 0 . (15)
1 −0 0 where the selective function f(2)(N)= N2+4N 3.
4 − −
The commutation relations among Q are the same
ij
In the U(3) case, the occupation number space is di- as those of eij, i.e., [Qij,Qkl] = δjkQil δliQkj.. All of
−
vided into four invariantsubspaces specified by the total the36Qij’scommutewiththetotalnumberoperator. If
particle number, which runs from 0 to 3. Generally, the Gi denotesthe fundamentalrepresentationofi-thgener-
occupation number space of U(n) group is divided into ator of the U(6) Lie group, then its Schwinger fermionic
n+1 invariant subspaces, where the total particle num- representation can be realized as
ber N runs from 0 to n, and the subspace with N = m
6,6
is conjugated to the subspace with N =n m.
− Gh = GαβQ
One may define a selective function i X i αβ
α=1,β=1
m−1 n−1 6,6
x i x i
fn(m)(x)= Y m− i Y m− i. (16) = X Oα†GαiβOβf4(2)(N). (20)
i=1 − i=m+1 − α=1,β=1
4
In general, one can construct the U(Cm) Lie algebra also obey the commutation relations of U(Cm) algebra.
n n
using n-fermion operators. To derive the mapping, we Therefore,whenm¯ =m,themoregeneralrepresentation
first define a set n-dimensional vectors ζ = (ζ ,ζ ...,ζ ) of U(Cm) are given6 by
1 2 n n
where each ζ = 0 or 1. There are 2n such vectors, and
i
Cm of them satisfy n ζ = m for a fixed integer m,
wnhere 0≤m≤n. LaPbei=l1thei Cnm vectors in the descend- Ghi =XGαiβQ(αmβ)ξ−+XG′iαβQα(m¯β)ξ+, (25)
ing order, e.g. the first one is αβ αβ
ζ(1) =(ζ1(1) =1,...,ζm(1) =1,ζm(1+)1 =0,...,ζn(1) =0).(21) where ξ± = 0 or 1, ξ++ξ− = 0, G and G′ denotes two
Cm Cm matrix representat6ions of the U(Cm) group.
Due to ζ(i), we can define a set of operators as n × n n
Oi =anζn(i)aζnn(−i−)11...a1ζ1(i). (22)
VI. CONCLUSION
The element operators are similarly defined as Q =
ij
O†O f(m)(N). Their commutation relations satisfy
i j n
[Q ,Q ] = δ Q δ Q . Then the Schwinger In conclusion, we have studied the non-standard
ij kl jk il li kj
fermionic representatio−nsofthe U(Cm) groupcanbe re- Schwinger fermionic representation of the unitary group
n
alized by by using n-fermion operators. we found that the
Schwinger fermionic representation of the U(n) group
Gm = GαβQm , (23) is not unique when n 3. In general, based on n-
i X i αβ ≥
fermion operators, the NSSFR of the U(n) group can
α,β
be established in a uniform approach,where all the gen-
where Gi is the i-th Cnm ×Cnm matrix generator of the erators commute with the total number operators. The
U(Cm) group, and Gαβ is its element in (α,β) position. Schwinger fermionic representation of U(Cm) group is
n i n
Letm¯ =n m,Cm¯ =Cm. ThesubspaceswithN =m also discussed.
− n n
and N = m¯ are mutually conjugate. The linear combi- We thank S. W. Hu for his valuable discussions. This
nations of the operators Qmi¯j work is supported in part by NSF of China (Grant No.
10575053andNo. 10605013)andProgramforNewCen-
Gm¯ = GαβQm¯ (24) tury Excellent Talents in University.
i X i αβ
α,β
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